Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-18T10:20:41.280Z Has data issue: false hasContentIssue false
  • English
  • Français

A Congruence Modulo Four for Real Schubert Calculus with Isotropic Flags

Published online by Cambridge University Press:  20 November 2018

Nickolas Hein ,
Frank Sottile  and
Igor Zelenko
Nickolas Hein
Affiliation:
Department of Mathematics and Computer Science, Benedictine College, Atchison, Kansas 66002, USA. nhein@benedictine.edu
Frank Sottile
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA. sottile@math.tamu.edu, zelenko@math.tamu.edu
Igor Zelenko
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA. sottile@math.tamu.edu, zelenko@math.tamu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

Keywords

14N1514P99Lagrangian GrassmannianWronski mapShapiro Conjecture
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 2 , 01 June 2017 , pp. 309 - 318
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Anderson, B. and Helmke, U., Counting critical formations on a line. SIAM J. Control Optim. 52(2014), no. 1, 219–242. http://dx.doi.org/10.1137/120890533 Google Scholar
[2] Eisenbud, D. and Harris, J., Divisors on general curves and cuspidal rational curves. Invent. Math. 74(1983), 371–418. http://dx.doi.org/10.1007/BF01394242 Google Scholar
[3] Eremenko, A. and Gabrielov, A., Degrees of real Wronski maps. Discrete Comput. Geom. 28(2002), no. 3, 331–347. http://dx.doi.org/10.1007/s00454-002-0735-x Google Scholar
[4] Eremenko, A. and Gabrielov, A., Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry. Ann. of Math. (2) 155(2002), no. 1,105-129. http://dx.doi.org/10.2307/3062151 Google Scholar
[5] Fehér, L. M. and Â. Matszangosz, K., Real solutions of a problem in enumerative geometry. Period. Math. Hungar. 73(2016), no. 2, 137–156. http://dx.doi.org/10.1007/s10998-016-0122-7 Google Scholar
[6] Finashin, S. and Kharlamov, V., Abundance of real lines on real projective hypersurfaces. Int. Math. Res. Notices (2012). Google Scholar
[7] Hein, N., Hillar, C. J., and Sottile, F., Lower bounds in real Schubert calculus. Sâo Paulo J. Math. Sci. 7(2013), no. 1, 33–58. http://dx.doi.org/10.11606/issn.2316-9028.v7i1p33-58 Google Scholar
[8] Hein, N. and Sottile, F., Beyond the Shapiro Conjecture and Eremenko-Gabrielov lower bounds, 2013, http://www.math.tamu.edu/∼Osecant/lowerBounds/lowerBounds.php. Google Scholar
[9] Hein, N., Sottile, F., and Zelenko, I., A congruence modulo four in real Schubert calculus. 2014, J. Reine Angew. Math., to appear.Google Scholar
[10] Itenberg, I. V., Kharlamov, V. M., and Shustin, E. I., Welschinger invariant and enumeration of real rational curves. Int. Math. Res. Not. (2003), no. 49, 2639–2653.Google Scholar
[11] Itenberg, I. V., Kharlamov, V. M., Logarithmic equivalence of the Welschinger and the Gromov-Witten invariants. Uspekhi Mat. Nauk 59(2004), no. 6(360), 85–110. http://dx.doi.org/10.4213/rm797 Google Scholar
[12] Kleiman, S. L., The transversality of a general translate. Compositio Math. 28(1974), 287–297. Google Scholar
[13] Martin del Campo, A. and Sottile, F., Experimentation in the Schubert calculus. In: Schubert Calculus (Osaka 2012), Advanced Studies in Pure Mathematics, 71, Mathematical Society of Japan, 2016, pp. 295–335.Google Scholar
[14] Mukhin, E. and Tarasov, V., Lower bounds for numbers of real solutions in problems of Schubert calculus. 2014, arxiv:1404.71 94.Google Scholar
[15] Mukhin, E., Tarasov, V., and Varchenko, A., The B. and Shapiro M. conjecture in real algebraic geometry and the Bethe ansatz. Ann. of Math. (2) 170(2009), no. 2, 863–881. http://dx.doi.org/10.4007/annals.2009.1 70.863 Google Scholar
[16] Mukhin, E., Schubert calculus and representations of the general linear group. J. Amer. Math. Soc. 22(2009), no. 4, 909–940. http://dx.doi.org/10.1090/S0894-0347-09-00640-7 Google Scholar
[17] Okonek, C. and Teleman, A., Intrinsic signs and lower bounds in real algebraic geometry J. Reine Angew. Math. 688(2014), 219–241.Google Scholar
[18] Okonek, C. and Teleman, A., A wall-crossing formula for degrees of real central projections. Int. J. Math. 25(2014), 1450038, 34 pp. http://dx.doi.org/! 0.1142/S01291 67X1 4500384 Google Scholar
[19] Purbhoo, K., Reality and transversality for Schubert calculus in OG(«, 2” + 1). Math. Res. Lett. 17(2010), no. 6, 1041–1046. http://dx.doi.org/10.4310/MRL.2010.v17.n6.a3 Google Scholar
[20] Soprunova, E. and Sottile, F., Lower bounds for real solutions to sparse polynomial systems. Adv. Math. 204(2006), no. 1, 116–151. http://dx.doi.org/10.1016/j.aim.2OO5.O5.O16 Google Scholar
[21] Sottile, F., The special Schubert calculus is real. Electronic Research Announcements of the AMS 5(1999), 35–39. http://dx.doi.org/10.1090/S1079-6762-99-00058-X Google Scholar
[22] Sottile, F., Some real and unreal enumerative geometry for flag manifolds. Mich. Math. J. 48(2000), 573–592. http://dx.doi.org/10.1307/mmjV1030132734 Google Scholar
[23] Sottile, F., General isotropic flags are general (for Grassmannian Schubert calculus). J. Algebraic Geom. 19(2010), no. 2,367-370. http://dx.doi.org/10.1090/S1056-3911-09-00518-9 Google Scholar
[24] Welschinger, J.-Y., Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry. C. R. Math. Acad. Sci. Paris 336(2003), no. 4, 341–344. http://dx.doi.org/!0.1016/S1631-073X(03)00059-1 Google Scholar